https://sheldonbrown.com/sturmey-archer/bsahub.html
which is great but it doesn't explain how the hub works or how it came to be.
I didn't realise this until recently but the design is actually derived from the Sturmey Archer 'X' series hub. The 'X' series hub is ancient; in SA production it was replaced by the 'K' series hub in 1921, so for most of its production run the BSA hub was already an antique, more or less.
So how does it work? The fundamental principle is like most other SA 3s hubs; in gears 1 and 2 the driver drives the ring gear, and the drive is taken to the hubshell from the planet cage or the ring gear respectively. In gear 3 the driver drives the planet cage and the hubshell is driven from the ring gear.
However the way this is achieved is quite unlike other SA hubs; there is no obvious sliding clutch, and the overall result is NIG type operation.
With reference to the figure below (which is from the Sheldon Brown link above)

Planet cage 'E' and Ring Gear 'F' constitute a subassembly and clip 'G' fits into F such that E is retained within F provided clip G is in place. This whole subassembly slides up and down the (enormously long) sun pinion and acts as the sliding clutch. There are three sets of pawls; part 'B' contains the low gear pawls (LGP); these operate somewhat like those in a K series hub. The other two sets of pawls are set in the RH end of part 'F', the ring gear. When the drive is taken from the ring gear to the hubshell (gears 2 and 3), the external high gear pawls 'S' (HGP) are used. The third set of (inwards facing) 'driver pawls' 'T' (DP) takes the drive from the driver to the ring gear, and are engaged in gear 1 and gear 2.
In the illustration you can see that the driver 'K' has some odd-shaped scallops in it; these are key to the operation of the gear, since they determine whether or not pawls DP and/or HGP can transfer drive or not.
- In gear 1 pawls DP engage with the driver but pawls HGP are defeated; either the scallops are not deep enough to allow pawls HGP to protrude, or (and I don't know which it is) the pawls cannot drive the dog ring 'V' because they are not yet far enough to the right . Drive is thus taken from the ring gear to the planet cage which then drives the hubshell via pawls LGP.
- In gear 2 pawls DP engage with the driver but in this position (with assy 'EFG' slightly rightwards) pawls HGP are allowed to engage with 'V' and give the direct drive gear 2. Pawls LGP would be overrun anyway, but at some point the movement of 'EFG' presumably disconnects the planet cage from these pawls.
- In gear 3 pawls DP are overrun and cannot transfer forwards drive. Meantime 'EFG' has slid rightwards far enough that the end of the driver 'K' engages with the planet cage directly, and pawls HGP are active (as in gear 2) so drive the hubshell at an increased speed vs the planet cage.
The hub ought to give good NIG performance, although the pawls don't change mode in a step change so much as gradually; this means that if the gear is out of adjustment pawl tips are likely to be damaged. If the hub is driven backwards in top gear, there is the same problem as is seen in the later NIG SA hubs, i.e. that both the ring gear and the planet cage are connected to the driver and they are moving at different speeds. In the BSA/'X' the end of the driver has reverse faces which are angled, so the planet cage is forced leftwards in the event of reverse torque between the driver and the planet cage. Since the gear selector key is spring-loaded in both directions, the required movement ought to be possible.
So all in all, it is an interesting hub. It should be a deal stronger than most SA hubs; the slotted part of the axle is well to the left of the hub (so won't see high loads from chain tension) and the driver appears to be supported by ball bearings at both ends, which ought to make for better-managed loads in the hub too. Probably it was discontinued by SA for the usual sort of reason, i.e. that it was slightly more expensive to make than the hub which succeeded it.
cheers